HOW TO WRITE EACH EXPRESSION AS A PRODUCT OF ITS FACTORS

List all the Factors of the monomial expression

Problem 1 :

Solution :

Decompose each expression into prime factors.

Finding the degree, coefficient and variables of following monomial expression

Variables :

The variables are the letters present in a monomial. That is, x, y, z,.. 

Coefficient :

The coefficient is the number that is multiplied by the variables. That is, 12x, 12y,..

Degree :

The degree is the sum of the exponents of the variables in a monomial.

That is, 12xy ---> Exponent of x is 1, Exponent of y is 1

So, the degree is 1 + 1 = 2

Problem 2 :

Solution :

Find the Greatest Common Factors of the following Monomial expression

Problem 3 :

12m²n³, 70m³n

Solution :

Find the GCF of 12m²n³, 70m³n.

12m²n³ = 2 ∙ 2 ∙ 3 ∙ m ∙ m ∙ n ∙ n ∙ n

70m³n = 2 ∙ 5 ∙ 7∙ m ∙ m ∙ m ∙ n

= 2 ∙ m ∙ m ∙ n

GCF = 2m²n

Problem 4 :

72a³b², 86a

Solution :

Find the GCF of 72a³b², 86a.

72a³b² = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 . a ∙ a ∙ a ∙ b ∙ b

86a = 2 ∙ 43 ∙ a

GCF = 2a

Problem 5 :

44m²n, 48mn²

Solution :

Find the GCF of 44m²n, 48mn².

44m²n = 2 ∙ 2 ∙ 11 ∙ m ∙ m ∙ n

48mn² = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ m ∙ n ∙ n

 = 2 ∙ 2 ∙ m ∙ n

GCF = 4mn

Problem 6 :

a²b³, ab³

Solution :

a²b³ = a ∙ a ∙ b ∙ b ∙ b

ab³ = a ∙ b ∙ b ∙ b

= a ∙ b ∙ b ∙ b

GCF = ab³

Problem 7 :

3x, 7xy²

Solution:

3x = 3 ∙ x

7xy² = 7 ∙ x ∙ y ∙ y 

GCF = x

Problem 8 :

4rs², 27st³

Solution:

4rs² = 2 ∙ 2 ∙ r ∙ s ∙ s

27st³ = 3 ∙ 3 ∙ 3 ∙ s ∙ t ∙ t ∙ t

GCF = s

Problem 9 :

18wx², 45wx

Solution:

18wx² = 2 ∙ 3 ∙ 3 ∙ w ∙ x ∙ x

45wx = 3 ∙ 3 ∙ 5 ∙ w ∙ x

= 3 ∙ 3 ∙ w ∙ x

GCF = 9wx

Problem 10 :

12y², 15y³, 5y

Solution:

12y² = 2 ∙ 2 ∙ 3 ∙ y ∙ y

15y³ = 3 ∙ 5 ∙ y ∙ y ∙ y

5y = 5 ∙ y

GCF = y

Problem 11 :

rs³, s³t, r²st²

Solution :

rs³ = r ∙ s ∙ s ∙ s

s³t = s ∙ s ∙ s ∙ t

r²st² = r ∙ r ∙ s ∙ t ∙ t

GCF = s

Problem 12 :

Find the GCF of the numbers.

a)  35, 56, 63

b)  30, 60, 78

c)  42, 70, 84

Solution :

a)  35, 56, 63

35 = 7 x 5

56 = 2 x 2 x 2 x 7

= 2³ x 7

63 = 3 x 3 x 7

=  3² x 7

Greatest common factor = 7

b)  30, 60, 78

30 = 2 x 3 x 5

60 = 2 x 2 x 3 x 5

78 = 2 x 3 x 13

Greatest common factor = 2 x 3

= 6

c)  42, 70, 84

42 = 2 x 3 x 7

70 = 2 x 5 x 7

84 = 2 x 2 x 3 x 7

Greatest common factor is 2 x 7, that is 14.

Problem 13 :

You are making balloon arrangements for a birthday party. There are 16 white balloons and 24 red balloons. Each arrangement must be identical. What is the greatest number of arrangements you can make using every balloon?

Solution :

Number of white balloons = 16

Number of red balloons = 24

16 = 2 x 2 x 2 x 2

= 2⁴

24 = 2 x 2 x 2 x 3

= 2³ x 3

Common factor = 2³

Greatest common factor of 16 and 24 is 8. So, in each arrangement we should use 8 balloons.

Problem 14 :

A science museum makes gift bags for students using 168 magnets, 48 robot figurines, and 24 packs of freeze-dried ice cream.

What is the greatest number of gift bags that can be made using all of the items?

How many of each item are in each gift bag?

Solution :

Number of magnets = 168

Number of robot figurines = 48

Number of freeze dried ice cream = 24

168 = 2 x 2 x 2 x 3 x 7

= 2³ x 3 x 7

48 = 2 x 2 x 2 x 2 x 3

= 2⁴ x 3

24 = 2 x 2 x 2 x 3

= 2³ x 3

Common factor = 2³ x 3

= 24

So, the greatest number of gift bags = 24. This means the maximum number of identical gift bags that can be made is 24. 

To find the number of each item per bag: 

• Magnets: 168 magnets / 24 bags = 7 magnets per bag
• Robot figurines: 48 figurines / 24 bags = 2 figurines per bag
• Freeze-dried ice cream: 24 packs / 24 bags = 1 pack per bag